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Quiz: Determinants and Matrix Properties

Test your understanding of determinants and fundamental matrix properties.

Note: This quiz covers key concepts from the chapter outline. Full chapter content is under development.

1. What does the determinant of a matrix represent geometrically?

  1. The sum of diagonal elements
  2. The signed volume scaling factor of the transformation
  3. The number of pivots in row echelon form
  4. The trace of the matrix
Show Answer

The correct answer is B. The determinant represents the signed scaling factor for volumes (areas in 2D). A determinant of 2 means the transformation doubles volumes; a negative determinant indicates the transformation reverses orientation.

Concept Tested: Determinant

2. If \(\det(A) = 0\), then the matrix \(A\) is:

  1. Orthogonal
  2. Symmetric
  3. Singular (non-invertible)
  4. Positive definite
Show Answer

The correct answer is C. A matrix with zero determinant is singular, meaning it has no inverse. The transformation collapses space in at least one dimension, making the operation irreversible.

Concept Tested: Singular Matrix

3. For any square matrices \(A\) and \(B\), which property holds?

  1. \(\det(A + B) = \det(A) + \det(B)\)
  2. \(\det(AB) = \det(A) \cdot \det(B)\)
  3. \(\det(AB) = \det(A) + \det(B)\)
  4. \(\det(A^{-1}) = \det(A)\)
Show Answer

The correct answer is B. The determinant is multiplicative: \(\det(AB) = \det(A) \cdot \det(B)\). This reflects that composing transformations multiplies their volume scaling factors.

Concept Tested: Determinant Properties

4. The trace of a matrix is:

  1. The product of diagonal elements
  2. The sum of diagonal elements
  3. The sum of all elements
  4. The determinant divided by the dimension
Show Answer

The correct answer is B. The trace of a square matrix is the sum of its diagonal elements: \(\text{tr}(A) = \sum_{i=1}^n A_{ii}\). The trace equals the sum of eigenvalues.

Concept Tested: Trace

5. What is \(\det(A^T)\) in terms of \(\det(A)\)?

  1. \(-\det(A)\)
  2. \(\det(A)\)
  3. \(1/\det(A)\)
  4. \(\det(A)^2\)
Show Answer

The correct answer is B. The determinant of a transpose equals the original determinant: \(\det(A^T) = \det(A)\). Transposing swaps rows and columns but doesn't change the volume scaling factor.

Concept Tested: Determinant of Transpose

6. If matrix \(A\) has determinant 5, what is \(\det(2A)\) for a \(3 \times 3\) matrix?

  1. 10
  2. 25
  3. 40
  4. 80
Show Answer

The correct answer is C. For an \(n \times n\) matrix, \(\det(cA) = c^n \det(A)\). For a \(3 \times 3\) matrix with \(\det(A) = 5\): \(\det(2A) = 2^3 \cdot 5 = 8 \cdot 5 = 40\).

Concept Tested: Scalar Multiplication and Determinants

7. A positive definite matrix has:

  1. All positive entries
  2. All positive eigenvalues
  3. Positive determinant only
  4. Positive trace only
Show Answer

The correct answer is B. A positive definite matrix has all positive eigenvalues. Equivalently, \(\mathbf{x}^T A \mathbf{x} > 0\) for all non-zero vectors \(\mathbf{x}\). This implies the matrix is invertible.

Concept Tested: Positive Definite

8. Which row operation does NOT change the determinant?

  1. Adding a multiple of one row to another
  2. Multiplying a row by a non-zero scalar
  3. Swapping two rows
  4. All row operations change the determinant
Show Answer

The correct answer is A. Adding a multiple of one row to another does not change the determinant. Swapping rows multiplies the determinant by \(-1\), and multiplying a row by scalar \(c\) multiplies the determinant by \(c\).

Concept Tested: Determinant and Row Operations

9. What is \(\det(A^{-1})\) in terms of \(\det(A)\)?

  1. \(\det(A)\)
  2. \(-\det(A)\)
  3. \(1/\det(A)\)
  4. \(\det(A)^2\)
Show Answer

The correct answer is C. Since \(AA^{-1} = I\) and \(\det(I) = 1\), we have \(\det(A) \cdot \det(A^{-1}) = 1\), so \(\det(A^{-1}) = 1/\det(A)\).

Concept Tested: Determinant of Inverse

10. A matrix is positive semidefinite if:

  1. All eigenvalues are strictly positive
  2. All eigenvalues are non-negative (zero or positive)
  3. The determinant is positive
  4. All entries are non-negative
Show Answer

The correct answer is B. A positive semidefinite matrix has all non-negative eigenvalues (zero or positive). Equivalently, \(\mathbf{x}^T A \mathbf{x} \geq 0\) for all vectors \(\mathbf{x}\).

Concept Tested: Positive Semidefinite